Difficult theoretical problem on basis vectors

[spaghetti3451]

How the hell do you prove that the components of a vector w.r.t. a given basis are unique?

I have literally no idea how to begin! It's just that with these theoretical problems there's no straightforward starting point!

 

[HallsofIvy]

Suppose they were not unique. That is, suppose there were some vector, v, such that
$$v= a_1v_1+ a_2v_2+ \cdot\cdot\cdot+ a_nv_n$$
and
$$v= b_1v_1+ b_2v_2+ \cdot\cdot\cdot+ b_nv_n$$
Now subtract:
$$(a_1- b_1)v_1+ (a_2- b_2)v_2+ \cdot\cdot\cdot+ (a_n- b_n)v_n= 0$$

One of the requirements for a "basis" is that the vectors are independent. Use the definition of "independent".

 

[spaghetti3451]

I see! If the basis vectors are independent, then the coefficients are zero, that is, the components are unique.

I have to tell you, though! It's not easy to figure out how to go about the problem.

I just wish the solution were as obvious as it looks now!

 

[dodo]

If it helps, the general pattern of any proof about "uniqueness" is to present two elements, and deduce that they are the same. The first two lines of Halls' proof are exactly that... and starting is usually the hardest thing. Once into it, you'll figure out a way of proving that your two elements are equal.

 

[blue_raver22]

I understand the frustration and difficulty in proving theoretical concepts. In order to prove the uniqueness of vector components with respect to a given basis, it is important to first define and understand the concept of a basis. A basis is a set of linearly independent vectors that span a vector space. This means that any vector in the vector space can be expressed as a linear combination of the basis vectors.

To prove the uniqueness of vector components, we can start by assuming that there are two sets of components for the same vector with respect to a given basis. Let's call these two sets of components A and B. Now, we can express the vector using both sets of components as:

v = A1b1 + A2b2 + ... + Anbn = B1b1 + B2b2 + ... + Bnbn

where A1, A2, ..., An and B1, B2, ..., Bn are the components of the vector with respect to the basis vectors b1, b2, ..., bn.

Since the basis vectors are linearly independent, we can equate the coefficients of each basis vector on both sides of the equation. This will give us n equations with n unknowns, which can be solved to show that A1 = B1, A2 = B2, ..., An = Bn. This proves that the components of a vector with respect to a given basis are unique.

In addition, we can also use the concept of linear independence to show that any other set of components for the same vector with respect to the given basis will also be the same as A and B. This further strengthens the uniqueness of vector components.

In conclusion, proving the uniqueness of vector components with respect to a given basis requires a deep understanding of the concept of a basis and the use of linear independence. It may also require the use of mathematical techniques such as solving equations to show the uniqueness of the components. I hope this explanation helps in tackling this difficult theoretical problem.